Jacobson radical of a module category and its power

Assume that k is a field and A is a k-algebra. The category of (left) A-modules will be denoted by Mod⁢(A) and HomA⁢(X,Y) will denote the set of all A-homomorphisms between A-modules X and Y. Of course HomA⁢(X,Y) is an A-module itself and EndA⁢(X)=HomA⁢(X,X) is a k-algebra (even A-algebra) with composition as a multiplication.

Let X and Y be A-modules. Define

radA⁢(X,Y)={f∈HomA⁢(X,Y)|∀g∈HomA⁢(Y,X) 1X-g⁢f⁢ is invertible in ⁢EndA⁢(X)}.

Properties.1) The Jacobson radical is an ideal in Mod⁢(A), i.e. for any X,Y,Z∈Mod⁢(A), for any f∈radA⁢(X,Y), any h∈HomA⁢(Y,Z) and any g∈HomA⁢(Z,X) we have h⁢f∈radA⁢(X,Z) and f⁢g∈radA⁢(Z,X). Additionaly radA⁢(X,Y) is an A-submodule of HomA⁢(X,Y).

2) For any A-module X we have radA⁢(X,X)=rad⁢(EndA⁢(X)), where on the right side we have the classical Jacobson radical.

In particular, if X and Y are not isomorphic, then radA⁢(X,Y)=HomA⁢(X,Y). □

Let n∈ℕ and let f∈HomA⁢(X,Y). Assume there is a sequence of A-modules X=X0,X1,…,Xn-1,Xn=Y and for any 0≤i≤n-1 we have an A-homomorphism fi∈radA⁢(Xi,Xi+1) such that f=fn-1⁢fn-2⁢⋯⁢f1⁢f0. Then we will say that f is n-factorizable through Jacobson radical.

Definition.The n-th power of a Jacobson radical of a category Mod⁢(A) is defined as a class

radn⁢Mod⁢(A)=⋃X,Y∈Mod⁢(A)radAn⁢(X,Y),

where radAn⁢(X,Y) is an A-submodule of HomA⁢(X,Y) generated by all homomorphisms n-factorizable through Jacobson radical. Additionaly define

radA∞⁢(X,Y)=⋂n=1∞radAn⁢(X,Y)⁢□

Properties.0) Obviously radA⁢(X,Y)=radA1⁢(X,Y) and for any n∈ℕ we have

radAn⁢(X,Y)⊇radA∞⁢(X,Y).

1) Of course each radAn⁢(X,Y) is an A-submodule of HomA⁢(X,Y) and we have following sequence of inclusions:

HomA⁢(X,Y)⊇radA1⁢(X,Y)⊇radA2⁢(X,Y)⊇radA3⁢(X,Y)⊇⋯

2) If both X and Y are finite dimensional, then there exists n∈ℕ such that